It is famously impossible to deduce the shape of a drum from its spectrum, in general. In the case of the hydrogen atom, there are non-Coulomb potentials that produce the same spectral series! (See https://doi.org/10.1103/PhysRevA.82.022121)
Is using the Coulomb potential only motivated by the correspondence to classical electrodynamics, or is there sufficient experimental information on the form of the orbitals to deduce the potential uniquely, assuming Schrödinger's equation?
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In principle, the quantum potential could differ from Coulomb only at very short lengthscales, leaving it compatible with the classical one. But having a different potential would impact the form of the orbitals!
In fact this is exactly the case, and it has exactly the effect that you describe.
The Coulomb potential is the most important contribution to the electron-proton interaction at low energies and long distances, such as those found in a hydrogen atom. However, the two particles also interact via the weak neutral current and (at second order) via the weak charged current. To the extent that the vacuum polarization around the electron contains virtual hadron loops, there's even a tiny bit of strong interaction contribution. If electrons and protons participate in some $\mathit{CP}$-violating interaction that hasn't been discovered yet (and we think that they do), that's hiding in the hydrogen atom as well.
When you "turn on" the weak interaction in your model, it changes the decay widths and allowed transitions in your mostly-Coulomb system: for instance, by allowing parity violation, even in "purely" electromagnetic transitions. I've mostly worked on parity-violating nuclear transitions, but I have plenty of colleagues who work on parity-violating transitions in atoms.
Without having read your linked paper (preprint here), I would tell you that, if the hydrogen atom were not built on the Coulomb potential, then the standard model of particle physics would have to include information about the corrections in order to give correct predictions in higher-energy experiments. Which it does: the additional interactions we have discovered so far are the weak and strong interactions.
